Abstract
In this article, we consider the problem of periodic homogenization of a Feller process generated by a pseudo-differential operator, the so-called Lévy-type process. Under the assumptions that the generator has rapidly periodically oscillating coefficients, and that it admits “small jumps” only (that is, the jump kernel has finite second moment), we prove that the appropriately centered and scaled process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the generator. The presented results generalize the classical and well-known results related to periodic homogenization of a diffusion process.
Highlights
The classical reaction–diffusion equation ∂t p(t, x) = b(x), ∇x p(t, x) +Tr c(x )∇x2 p(t, x)r p(t, x) describes the evolution of population density due to random displacement of individuals, movement of individuals within the environment, and their reproduction
In order to characterize long-range effects the diffusion and drift terms are naturally replaced by an integro-differential operator of the following form
We focus to the case when {Xt }t≥0 is a so-called Lévy-type process or, equivalently, when L is a pseudo-differential operator
Summary
Let {Xt }t≥0 be a d-dimensional LTP with semigroup {Pt }t≥0, symbol q(x, ξ ) and Lévy triplet (b(x), c(x), ν(x, dy)), satisfying (C1), (C2), (C3) and (C4) x → b∗(x) := b(x) + B1c(0) y ν(x, dy) is of class Cbψ (Rd ) for some Hölder exponent ψ(r ), and (i) for some t0 > 0, any t ∈ (0, t0] and any τ -periodic f ∈ Cb(Rd ), Pt f ψ ≤ C(t) f ∞, where t0 0 In the following proposition we slightly generalize [63, Lemma 4.2] (see [27]), and prove Itô’s formula for a pure-jump LTP with respect to a not necessarily twice continuously differentiable function.
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